Just like the title says. Monks can't use Flurry of Blows unless unarmored. Is there a way around this? Feat, some other class that can use Flurry of Blows but isn't subject to the same restriction?
“Flurry of Blows”
The Shou disciple prestige class from Unapproachable East allows flurry of blows to be used in light armor. Medium or heavy armor, or using a shield, still interferes with it. The Shou disciple also gains the ability to flurry with any light melee weapon, rather than only with monk weapons, though this isn’t particularly impressive.
The exotic weapon master prestige class from Complete Warrior offers a “flurry of strikes” option. This only adds a single attack, and always applies a −2 penalty (i.e. it is equivalent to a 1st-level monk’s flurry of blows), but it has no armor restrictions. It must be used with an exotic double weapon or with a spiked chain.
The Defenders of the Faith version of the sacred fist prestige class gets a “flurry attack” feature that allows an extra attack for a −2 penalty, with no other requirements, but Defenders of the Faith is a 3.0e book and the 3.5e version of the sacred fist, in Complete Divine, does not offer this ability.
The tribal protector prestige class from Sword and Fist gets a “wild fighting” feature that trades a −2 attack penalty for an extra attack, with no other requirements. Mentioned because it is explicitly “[s]imilar to a monk attacking with a flurry of blows,” which of course is an accurate description.
Other ways to get more attacks
There are many ways to get more attacks. The haste spell, and the related speed special weapon property, are the most famous examples. The swiftblade prestige class deserves special mention just for doing so much more with haste.
Not quite the same thing, but it’d be remiss to leave Two-Weapon Fighting et al. out here, since those also can give more attacks for a −2 penalty on all attacks. Having to use two separate weapons, having to use one-handed and light weapons, having to take numerous feats, are all significant drawbacks, however.
The whirling frenzy rage variant is another, easy source of an extra attack, available to anyone who dips barbarian—which is already a good idea thanks to Complete Champion’s lion spirit totem option, which gives pounce.
Finally, worth mentioning is the Snap Kick feat from Tome of Battle: with this, you take a −2 penalty on attacks to make an extra unarmed strike, but what’s special about is—unlike almost all other options—you can use it any time you would make one or more attacks. All the other options in this answer require a full-attack, which is very limiting since you cannot move, and also cannot use some combat features that require a standard action (such as the strike maneuvers from the same Tome of Battle).
Attempting to list every single way to get extra attacks seems beyond the scope of the question.
Dark Sun, Athas has a couple of more possible options.
Platons: a life shaped graft that adds to natural armor bonus, and state:
A platon grafts onto a part of your body and acts like plate armor. Because platons become a part of your body, you do not suffer the penalties for such armor.
Which should count as unarmored, despite the description, depending on the DM.
Submitting to the manipulations of a Lifeshaping Mastershaper to replace limbs can result in masterwork limbs. Which could then technically be enhanced directly with psionics or spells as if armor... or as a weapon, in the case of a monk. A lifeshaped creature could be made entirely masterwork, from head to toes.
This is the only known method of making a monk character's body gain the masterwork quality that I am aware of in the game.
Note that lifeshaped items have their own set of rules, are neither magical nor psionic, and require expensive care and feeding (yes, they are living things).
Also, the costs for lifeshaped items in the current version of the 3.5 Dark Sun rules are somewhat skewed (especially for entire creatures) compared to the costs of similar effects elsewhere, so some DMs may not allow them. These costs may have been calculated based on the basis of expensive care and feeding combined with the fact that the items and creatures all die within a few years, necessitating periodic replacement.