You probably know this, but cherry and mahogany are dark reddish-brown (mahogany being darker), whilst oak is light yellowish. Lighter wood makes a room look larger, whilst the darker wood gives it a warmer, richer ambiance, so your room size and "type" must play an important role in the decision.
Of the three woods, the Mahogany will be the strongest. The Oak is rather difficult to work with (ask me, I've built my tank enclosure from solid Oak...), and the Cherry wood will most likely be the most expensive. I think Cherry is the most beautiful of the three, but you need a large "old" room to do it justice.
Good decision - there's nothing like solid wood, if you can afford it. Are you going to do the work yourself? If so, and you're not very experienced, I would concur with Kanga's recommendation of Beech, it is smooth-grained, and easy to work with. If you are more experienced, or if you are going to have it made, I would go for either the Oak or the Cherry, depending on your room.
Hennie