Is there a playable race of undead in Dungeons and Dragons 3.5? I would prefer a race that isn't a vampire or lich. If possible, please share their stats.
Yes. The best "playable" undead race is the necropolitan, found in (I believe) Libris Mortis. This template is the best choice for a PC because it has no level adjustment, merely causing the character in question to lose a level. Therefore, any build is possible with this basis, with especial attention being paid to the taint rules from Heroes of Horror.
Beyond that, look in Savage species for a way to make templates for any creature of your choice and many interesting exotic (and some undead) races. Libris Mortis also has some interesting undead rules.
If you take the D&D 3e setting of Ghostwalk you've got the option to play a ghost as well. Most of 3e's balance issues came with classes, so there shouldn't be any concern of Ghostwalk creating balance problems in 3.5.
The Unbodied while being a psionic race, could also be easily used to handle a ghost.
Another thing you could do is basically recreate the 4e Revenant. Give it undead status, and all the benefits that go with it, including no constitution score, and if you really want to give it Chill Touch as an innate ability, and have it come out at +1 to +2 levels (I'm thinking +1 with just undead status, +2 with chill touch as being able to repeatedly hammer in strength damage while unarmed is pretty powerful in the long run, as would converting a wizard to d12 hit dice).
Although World of Warcraft d20 is not DnD 3.5 in the strict sense, it uses basically the same rules (with some touches from Arcana Unearthed, like race levels) so the Forsaken can be cut and pasted without problems from one RPG to the other.
There are many playable undead in 3.5, even discounting vampires and liches; however, the most accessible is without a doubt creatures with the necropolitan template, from Libris Mortis.
I do not have that book on me, but even if I did it would proooobably not be okay to just transcribe those stats. You're on your own for that.
To give you a kind of empowering and loose answer: Just pick any undead from any monster supplement you have access to, get the approval of / discuss the details (level adjustment, background, house-rule tweaks concerning stats, skills etc) with your DM and there you go. ;)
To get you started, check out the Monsters as Races section of the d20srd.
There was a dragon article about playing half-undead that was pretty cool. Some were a bit silly, but for the most part it was well written and the classes seemed fairly balanced.
Under normal circumstances, crossbreeding the living with the undead is impossible. Necromancy and accidents of fate, however, can produce extremely unusual mixtures at life and unlife.
Creatures touched with the taint at undeath come in two forms: half-undead and deathtouched. Half-undead are direct descendants of undead creatures, while deathtouched are generations removed from their undead progenitors.
Both sorts at creatures are presented below for DMs and players who would like to breathe a bit more unlife into their campaigns. A half-undead template is offered for each of the most common kinds of half-undead, and the mortif deathtouched is presented as a new monster, similar to the planetouched, with the necessary information for use as a player character.
Fiendish Codex II: Tyrants of the Nine Hells, page 77, has a Race called 'Hellbred' that is very technically undead, and since their appearance is a curse given to them by a Devil, it is not outside of the realm of possibility that the appearance may be skeletal or otherwise undead in nature.
+2 Charisma and -2 Constitution If you choose the Soul(Su) aspect
+2 Constitution and -2 Intelligence if you choose the Body(ex) aspect
As a racial ability they can also use weapons and spells regardless of alignment restrictions.
Is a decent race and has good Roleplay-ability and a little less hassle then the templates or monster classes in Libris Mortis.
There was a Dragon magazine article (I can't find it at the moment) a few years back about playing undead characters as PCs and I actually got to make use of it. I had a character in a campaign who was killed by a bodak. As per the rules, 24 hours later he returned as a bodak himself, but through DM/divine intervention still retained enough of himself that he joined the party as a bodak character. He eventually died because the cleric of the party refused to channel negative energy to heal him after combats, despite knowing that the bodak was at least partly his former companion. It turned out pretty cool, and well played in character.