Getting just the terms independent of a certain variable

I have a very long and convoluted expression with many variables and my goal is to get just the terms which are coefficients of $e^{it}$. The variable $t$ never appears outside of exponential terms (only as $e^{ikt}$ for $k\in\mathbb{Z}$), so when I multiply through by $e^{-it}$, the terms I'm looking for are independent of $t$, while every other term has some $t$ dependence in it.

Is there a way to return only the terms which are independent of $t$ (while being dependent on some other variables)?

Coefficient would work in this case.

expr = E^(I Range[5] t). RandomInteger[{1, 10}, {5, 1}] // First


2 E^(I t) + 6 E^(2 I t) + 5 E^(3 I t) + 10 E^(4 I t) + E^(5 I t)

then

Coefficient[expr, E^( I t)]


2

This related question gives an answer to my own. The first comment suggests:

Select[expr,FreeQ[t]]


This returns the expression given but removes all terms which are dependent on the variable $t$.

You could try using FourierCoefficient. For example:

expr = x Exp[I t] + 2 y + 3 z Exp[-I t];
FourierCoefficient[expr, t, 1]


x