I wrote a good guide for Ryze detailing how to play him, what to build and why. Check it out and let me know what you think.

http://solomid.net/guides.php?g=606

Personally, i have try a lot of build for the new ryze, but the best thing i found was that:

1- Ionian Boot of Lucidity or Sorcerer Shoes
2- Archangle
3- Frozen Heart (For tank more Turret/Minion and Cooldown Reduction)
4- Banshee (Only if they have 2 Big Caster or a lot of stun/slow)
4.1- Rod Of Age insteed of Banshee
5- Will of the Ancients (Ryze is probably the only caster who can feel up all his hp in 5 sec with 20% Spell vamp, every 10 sec)
6- An item depending the ennemy formation. (Abyssal Scepter , Zhonya's Hourglass, Void Staff, Guardian Angel)
6.1- Hextech Gunblade (ONLY if you are always with a friend who have Will Of The Ancients too, Aura item can stack 2 time max on each champ, so u will have 60% Spell Vamp who make u invincible and impossible to lost HP cause 1 AoE spell with Ryze's ult= 90% Spell Vamp for 2 targets, 120% for 3, 150% for 4 and this is only for 1 Spell)

This is a good combination between Dommage, Armor, HP and Good HP Recovery.

RoA
AA
Banshees
Frozen Heart
Merc Treads
Lol as you solo teams with your ridiculous survivability.

Final item can be like whatever the **** you want honestly. Abyssal/WoTA for team aura as MR reduce is great and WoTA is good with your ultimate. You live forever!

tear, merc treads, BV, FH, FoN, Omen/thormail

then, the part you are all missing...

Tear -> manamune. 200 AD.

I assure you, the AD scales better than the AP. Also, I guess if they ALL stack tons of MR, you have to get Void... so probably drop FoN or omen, depending on team comp.

Boots 2, Archangels staff, Frozenheart, Rod of Ages, Banshee's veil, Void staff. Usual endgame build. Gives you about 3k health, 4k+ mana andabout 320ish ability power with 30% CD. (masteries or runes not included)

As for the AP versus mana debate, Mana is a far superior item for Ryze dps.

For a tanky Ryze I go with 9 - 7 - 14 for my masteries so that I can get a full 1% of my max mana as HP regen (from Defensive tree). With 4,000+ mana end game, you're getting an extra 8HP back every second and that can make a big difference dancing in and out of team fights.

I usually get Magic Pen, Move Speed quints, and Mana per Level Seals and Glyphs. (this gives 400 extra mana by level 18 and scales up damage, healing, and spammability.

I like turning the tear into an archangels staff, but, Manamune would allow him to make a bursty-item like sheen fairly good for Ryze too.

Rod of Ages are good on Ryze of course.

I try to get over 100 MR and Armor, with over 3000 HP.

Magic Pen boots are good but one of the problems with Ryze is that he has short ranges and can be kited if he doesn't have enough move speed.

Late game, Ryzes combos are insane and he's very, very solid. I get a lot of assists as if I was a tank, just because everyone focuses the guy I snare and combo.

Don't listen to LilSam23

RoA is amazing on Ryze especially when purchased early on. At max stats its one of the best items for the cost.

I almost always start with Sapphire Crystal and one of each potion. B when I get enough for catalyst. Some go tear first but I like the extra survivability catalyst gives early on plus the hp and mana per level allows you to stay in lane longer. After that I get sorc boots, finish roa and then start on either Frozen Heart if majority of other team is AD or Banshee's if AP or a lot of disables.

Ryze builds the same way in his "new" form as he did in his previous incarnation, except now the build performs better due to his changes. He's always been high mana + high survival because of his short range and really short cooldowns, which keeps him near danger longer than most other casters. Setting him up for "assisting the team in kills" will give you the same build as one designed to maximize his destructive potential as the only items you'll have for AP are an AA Staff, RoA and Abyssal Sceptre. You might include a Void Staff but I personally don't like it. The rest of your items are survival with an emphasis on +Mana. As you can see, even the AA Staff and RoA are chosen to take advantage of the bonus damage from Mana, and the Abyssal is there because you should always be in range to use the aura and the MR reduction stacks nicely with your E bounces.

Boots (Sorc or Merc, depending)
Banshee's Veil
RoA
AA Staff
Abyssal
Glacial Shroud (Frozen Heart if you have leftover gold)

That's pretty much his build regardless of the game. You can sub out Abyssal for a Void Staff if 3 or more priority targets are stacking MR to counter you, but again, I don't like the item. The hard part is building the items in the best order for the game. Usually getting an early Tear and Catalyst with Boots1 is the start you want to shoot for. You can build a second Catalyst after that or finish BV first, it really doesn't matter. RoA comes next, then you should probably pick up at least a Chain Vest before doing anything else. Converting the Tear into an AA Staff can be done whenever you think you can spare the gold, but it's honestly not a huge priority for Ryze.