The length of the plastic barrel part is what modifies the frequency response. The grill on the end only provides protection. You can use the microphone with just the barrel portion without a gill and retain the normal sonic character of the microphone as using an undamaged grid, as long as it is installed so that the solid-walled part extends over the capsule by the same length as a normal, undamaged grid.
When in a pinch, I've also modified the longer 'high-boost' grids (which I never use) to make 'low-boost' short grids out of them. To do that you need to pop out the grill part and remove it using a chopstick or something, then sand-down the longer tube so that it is the length as a short one.
You can then record normally until you can get replacements. The low-boost +3dB 'short-grids' are mostly likey the ones you want to specify when ordering replacements.
Some would say that 4061's sound even better without the grill because you're not getting the high frequency bump.
Not me. Using the microphone without a short-grid will eliminate the slight ~3dB boost centered around @10kHz, which more times than not is beneficial, creating something a diffuse-field equalized response- a useful trait in omnis intended for recording from a distance.
But not using the short-grid will also make the response somewhat less linear, at least to my ear and from the response graphs I've seen which show larger than -3dB wiggles.
[Edit- Correction. According to the graph I came across previously and posted at the link aaronji added in his post below, I was imagining those less-linear deviations. Looks like not using the short grid produces about the same response up to 10kHz where it's about +2 dB, but it then rolls back off just above that, instead of extending to +3dB @ 15kHz and then rolling off above there.] If the resulting recording does happen to be a bit over-energetic around 10kHz, a well-tuned minor EQ adjustment can fix that more effectively IME.
The longer, high-boost grids (the ones with the non-protruding screens) are intended to correct for a muffled response from on-person-mounted voice mic'ing, and produce a much peakier +10dB boost with a steeper drop-off way up top. I've never found those useful at all for recording music.