Filter odd or even harmonics with notch or inverse notch filter
What you are looking for are what we, in the audio space, call comb filters.  Comb filters may or may not have a feedback path, just like FIR and IIR filters. In fact, there is a generalized theory about designing comb filters based simply on designing any digital filter.
In fact, you can think of every LTI digital filter as being a comb filter where the teeth of the comb are spaced apart by the sampling frequency.  To space the teeth apart by a frequency spacing much less than the sampling frequency, you need a delay element that is longer than a single-sample delay ($z^{-1}$).  If a delay with precision of an integer number of samples is good enough to tune your comb filter, good for you.  But if it isn't, you need to deal with interpolation and precision delay.