How to filter uniform noise in a microcontroller?
You need to check the noise bandwidth using an oscilloscope. By saying noise I mean the hardware noises before digitalization by ADC. 
If the noises have high frequency component, remove it by analog filters. High frequency is defined as frequency higher than 1/2 of the sampling rate. And in practice, usually a sampling rate of 8-10 times of the pass band of the analog low pass filter is used. You may try simple RC filters.
Use FIR in a MCU to manipulate your data. Depends on the speed requirement, you can choose number of taps to use and etc. You should be able to find libraries provided by your chip manufacture, since FIR is very common in use. 


Hope it helps!