Three ancient poems were created as mnemonic aids for remembering the rune symbols, their names, meanings and properties. Because the runes changed slightly in each culture the three poems differ in some aspects.

From Runic and Heroic Poems by Bruce Dickins

Fe
Wealth is a source of discord among kinsmen;
the wolf lives in the forest.

Ur
Dross comes from bad iron;
the reindeer often races over the frozen snow.

Thurs
Giant causes anguish to women;
misfortune makes few men cheerful.

As
Estuary is the way of most journeys;
but a scabbard is of swords.

Reidh
Riding is said to be the worst thing for horses;
Reginn forged the finest sword.

Kaun
Ulcer is fatal to children;
death makes a corpse pale.

Hagall
Hail is the coldest of grain;
Christ created the world of old.

Naudhr
Constraint gives scant choice;
a naked man is chilled by the frost.

Isa
Ice we call the broad bridge;
the blind man must be led.

Ar
Plenty is a boon to men;
I say that Frodi was generous.

Sol
Sun is the light of the world;
I bow to the divine decree.

Tyr
Tyr is a one-handed god;
often has the smith to blow.

Bjarkan
Birch has the greenest leaves of any shrub;
Loki was fortunate in his deceit.

Madhr
Man is an augmentation of the dust;
great is the claw of the hawk.

Logr
A waterfall is a River which falls from a mountain-side;
but ornaments are of gold.

Yr
Yew is the greenest of trees in winter;
it is wont to crackle when it burns.

Norwegian Rune Poem in Old Norse (dead link)

(in Modern English)
From Runic and Heroic Poems by Bruce Dickins

Fé - Wealth
Source of discord among kinsmen
and fire of the sea
and path of the serpent.

Úr - Shower
Lamentation of the clouds
and ruin of the hay-harvest
and abomination of the shepherd.

Thurs - Giant
Torture of women
and cliff-dweller
and husband of a giantess.

Óss - God
Aged Gautr
and prince of Ásgardr
and lord of Vallhalla.

Reid - Riding
Joy of the horsemen
and speedy journey
and toil of the steed.

Kaun - Ulcer
Disease fatal to children
and painful spot
and abode of mortification.

Hagall - Hail
Cold grain
and shower of sleet
and sickness of serpents.

Naud - Constraint
Grief of the bond-maid
and state of oppression
and toilsome work.

Iss - Ice
Bark of rivers
and roof of the wave
and destruction of the doomed.

Ár - Plenty
Boon to men
and good summer
and thriving crops.

Sól - Sun
Shield of the clouds
and shining ray
and destroyer of ice.

Tyr
God with one hand
and leavings of the wolf
and prince of temples.

Bjarken - Birch
Leafy twig
and little tree
and fresh young shrub.

Madr - Man
Delight of man
and augmentation of the earth
and adorner of ships.

Lögr - Water
Eddying stream
and broad geysir
and land of the fish.

Yr - Yew
Bent bow
and brittle iron
and giant of the arrow.

Icelandic Rune Poem (in Old Icelandic)

(in Modern English)
From : Runic and Heroic Poems, by Bruce Dickins.

Feoh
Wealth is a comfort to all men;
yet must every man bestow it freely,
if he wish to gain honour in the sight of the Lord.

Ur
The aurochs is proud and has great horns;
it is a very savage beast and fights with its horns;
a great ranger of the moors, it is a creature of mettle.

Thorn
The thorn is exceedingly sharp,
an evil thing for any knight to touch,
uncommonly severe on all who sit among them.

Os
The mouth is the source of all language,
a pillar of wisdom and a comfort to wise men,
a blessing and a joy to every knight.

Rad
Riding seems easy to every warrior while he is indoors
and very courageous to him who traverses the high-roads
on the back of a stout horse.

Cen
The torch is known to every living man by its pale, bright flame;
it always burns where princes sit within.

Gyfu
Generosity brings credit and honour, which support one's dignity;
it furnishes help and subsistence
to all broken men who are devoid of aught else.

Wynn
Bliss he enjoys who knows not suffering, sorrow nor anxiety,
and has prosperity and happiness and a good enough house.

Haegl
Hail is the whitest of grain;
it is whirled from the vault of heaven
and is tossed about by gusts of wind
and then it melts into water.

Nyd
Trouble is oppressive to the heart;
yet often it proves a source of help and salvation
to the children of men, to everyone who heeds it betimes.

Is
Ice is very cold and immeasurably slippery;
it glistens as clear as glass and most like to gems;
it is a floor wrought by the frost, fair to look upon.

Ger
Summer is a joy to men, when God, the holy King of Heaven,
suffers the earth to bring forth shining fruits
for rich and poor alike.

Eoh
The yew is a tree with rough bark,
hard and fast in the earth, supported by its roots,
a guardian of flame and a joy upon an estate.

Peordh
Peorth is a source of recreation and amusement to the great,
where warriors sit blithely together in the banqueting-hall.

Eolh
The Eolh-sedge is mostly to be found in a marsh;
it grows in the water and makes a ghastly wound,
covering with blood every warrior who touches it.

Sigel
The sun is ever a joy in the hopes of seafarers
when they journey away over the fishes' bath,
until the courser of the deep bears them to land.

Tir
Tiw is a guiding star; well does it keep faith with princes;
it is ever on its course over the mists of night and never fails.

Beorc
The poplar bears no fruit; yet without seed it brings forth suckers,
for it is generated from its leaves.
Splendid are its branches and gloriously adorned
its lofty crown which reaches to the skies.

Eh
The horse is a joy to princes in the presence of warriors.
A steed in the pride of its hoofs,
when rich men on horseback bandy words about it;
and it is ever a source of comfort to the restless.

Mann
The joyous man is dear to his kinsmen;
yet every man is doomed to fail his fellow,
since the Lord by his decree will commit the vile carrion to the earth.

Lagu
The ocean seems interminable to men,
if they venture on the rolling bark
and the waves of the sea terrify them
and the courser of the deep heed not its bridle.

Ing
Ing was first seen by men among the East-Danes,
till, followed by his chariot,
he departed eastwards over the waves.
So the Heardingas named the hero.

Ethel
An estate is very dear to every man,
if he can enjoy there in his house
whatever is right and proper in constant prosperity.

Dæg
Day, the glorious light of the Creator, is sent by the Lord;
it is beloved of men, a source of hope and happiness to rich and poor,
and of service to all.

Ac
The oak fattens the flesh of pigs for the children of men.
Often it traverses the gannet's bath,
and the ocean proves whether the oak keeps faith
in honourable fashion.

Æsc
The ash is exceedingly high and precious to men.
With its sturdy trunk it offers a stubborn resistance,
though attacked by many a man.

Yr
Yr is a source of joy and honour to every prince and knight;
it looks well on a horse and is a reliable equipment for a journey.

Ior
Iar is a river fish and yet it always feeds on land;
it has a fair abode encompassed by water, where it lives in happiness.

Ear
The grave is horrible to every knight,
when the corpse quickly begins to cool
and is laid in the bosom of the dark earth.
Prosperity declines, happiness passes away
and covenants are broken.

Anglo-Saxon Rune Poem (in Anglo-Saxon)

There is yet another rune poem, the Abcedarium Nordmanicum, discovered in a manuscript written in the 9th century, written in high and low German. This is sometimes referred to as the "Old Swiss Rune Poem". It relates to the Younger Futhark.

Fee first,
Aurochs after,
Thurs the third stave,
The Åse is above him,
Wheel is written last,
Then cleaves cancre;
Hail has need;
Ice, year, and sun.
Tiu, birch and man in the middle;
Water the bright,
Yew holds all.

Three versions of the Abcedarium Nordmanicum are available on the Woden's Harrow website, along with a Real Audio sound file.