Edit: Note that these are unofficial fixes, meant more for the whole group/campaign than a single character. There is no official way that I know of, besides Vow of Poverty, which has its own balance issues.
In the past, I have changed the level progression slightly to include static bonuses to replace the core set of magic items (stat boosters, magic weapons/armor/shields, cloaks of resistance). This is unofficial / homebrew, but it is relatively well balanced, and follows normal equipment progression. It gives players something plenty to look forward to when leveling. It has proven relatively successful (for me at least).
- Feats every odd level (1st, 3rd, 5th, 7th...)
- Ability Score increases every even level (2nd, 4th, 6th, 8th...)
- +1 to Attack and Damage every four levels (3rd, 7th, 11th 15th...)
- +1 to Armor and Shield AC every four levels (4th, 8th, 12th, 16th...)
- +1 to all Saving Throws every four levels (5th, 9th, 13th, 15th...)
The Midnight OGL role-playing game by Fantasy Flight Games has a very cool mechanic called Heroic Paths. These are chosen in addition to a character's class, and grants them spell-like abilities that level along with them.
This could help replace the multitude of options and abilities granted by magic items.