Mighty Deeds are one of my favorite bits in DCC, but I sometimes feel that if you have this elegant way of resolving everything that might be a maneuver in a clunkier system (push back, knock down, disarm, blind, feint, etc.) it would be nice if you could apply that to 0-level characters and other classes trying the same kind of stunts. You just wouldn’t want them to be as good at it, since making Warriors cool and special is part of the point.
Thus, the following house rule: anybody can at least try a minimal deed in conjunction with their regular action (called here a Paltry Deed) by rolling a d6. Whatever they roll isn’t added to their to-hit or damage, but if they get a 6 and their Action Die is also a success then they accomplish their declared Deed as if they’d the rolled the lowest level of a Mighty Deed.
Variant 1: You don’t want them trying all the time, so to attempt a Paltry Deed they have to drop their action die 1 level.
Variant 2: That’s all very well, but shouldn’t they be slightly better if they have a high attribute value in the thing they’re attempting? Well, the attribute value will help them in the roll on the Action Die, but if you want to tie it to Attribute Bonus then they have to get the highest result possible on a :
| Paltry Deed succeeds on highest possible roll on | |
|---|---|
| -3 | d12 |
| -2 | d10 |
| -1 | d8 |
| 0 | d6 |
| +1 | d5 |
| +2 | d4 |
| +3 | d3 |
Variant 3: The real problem with Paltry Deeds is you never improve as you level up, except insofar as your attack bonus improves your chance of a success on the Action. OK, then look at the Attack Bonus for your level:
| Paltry Deed succeeds on highest possible roll on | |
|---|---|
| 0 | d6 |
| 1-2 | d5 |
| 3-4 | d4 |
| 4-5 | d3 |
| 6+ | d2 |
Variant 4: Some wacky combination of the above. Left as an exercise for the reader.
Tales of the Rambling Bumblers 
Good idea. Does the fighter get other abilities besides Mighty Deeds in DCC, I forget?
Yes, besides the deed die also being a to-hit and damage bonus, fighters get better and better crit dice/tables, they’re the only class whose init improves, and starting at 5th level they get an improving 2nd action die. Plus they get d12 for hit dice.