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Question about DiracDelta

Hi group, I need to plot this:

Plot[NIntegrate[DiracDelta[3 - Cos[x] - Cos[y] - Cos[z] - e], {x, -Pi,
Pi}, {y, -Pi, Pi}, {z, -Pi, Pi}, MaxPoints -> 1000000]/8/Pi^3, {e, 0,
6}, PlotPoints -> 30]

the problem is that the function always returns 0, no matter what the
value of 'e' is. I saw that other people have reported incorect
evaluation/simplification related to DiracDelta, but I was wondering
can anyone tell me if there's some workaround?

Thanks in advance

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