mjhjmtu

I think this is what you want:

IntegerDigits[35 - 1, MixedRadix[5, 4, 3], 3] + 1

In general:

gridIndex[n_Integer, shape_List] := 
 IntegerDigits[n - 1, MixedRadix[shape], Length@shape] + 1

If you have enough memory, then a lookup table may be fastest:

shape = 5, 4, 3;
indices = Tuples[Range /@ shape];

Lookup is fast:

indices[[35]] // RepeatedTiming
(* 3.*10^-7, 3, 4, 2 *)

Also, it seems that doing lots of lookups simultaneously is even faster (per lookup):

indices[[22, 45, 35, 49, 36, 9, 9, 39, 59, 14]] // RepeatedTiming

Here is the obligatory compiled version; it is not as fast as Roman's lookup method but it is also less memory hungry (in particular if indexing is supposed to be done into SparseArray whose dense version does not fit into memory).

A compiled helper function:

cf = Compile[n, _Integer, mods, _Integer, 1,
 Block[r = n - 1, d, m,
 Table[
 m = Compile`GetElement[mods, i];
 d = Quotient[r, m];
 r = r - d m;
 d + 1,
 i, 1, Length[mods]]
 ],
 CompilationTarget -> "C",
 RuntimeAttributes -> Listable,
 Parallelization -> True,
 RuntimeOptions -> "Speed"
 ];
getInds[idx_, shape_] :=
 cf[idx, Reverse[Most[FoldList[Times, 1, Reverse[shape]]]]]

Usage example and timing test, comparing against swish's and Roman's proposals:

RandomSeed[123];
shape = RandomInteger[1, 10, 4];
idx = RandomInteger[1, Times @@ shape, 10000];

a = gridIndex[#, shape] & /@ idx; // RepeatedTiming // First

b = getInds[idx, shape]; // RepeatedTiming // First

indices = Tuples[Range /@ shape];
c = indices[[idx]]; // RepeatedTiming // First
a == b == c

0.429

0.00059

0.0000700

True

Here's what I came up with:

getSubindex[index_, stride_] := 
 Mod[index, stride, 1],
 Ceiling[index/stride]
 
getIndex[index_, strides_] := 
 Reverse@FoldPairList[getSubindex, index, Reverse@strides]

This is comparable to swish's solution speed-wise:

gridIndex[1000, 3, 5, 4, 6] // RepeatedTiming

0.000061, 3, 2, 3, 4

getIndex[1000, 3, 5, 4, 6] // RepeatedTiming

0.000052, 3, 2, 3, 4

So after I posted the question last night I came up with a solution that is fast and vectorized: (note that if you're working with huge numbers you'll need to remove the N for accuracy, but you'll incur a huge speed penalty)

gifs[inds_, strides : __Integer] :=
 Module[
 
 accstr,
 stride = strides,
 ind = inds - 1,
 moddable,
 modres
 ,
 accstr =
 N@
 Append[
 Reverse@FoldList[Times, strides[[-1 ;; 2 ;; -1]]],
 1
 ];
 moddable = If[ListQ@inds, Map[ind/# &, accstr], ind/accstr];
 modres = 1 + Mod[Floor[moddable], stride];
 If[ListQ@inds, Transpose, Identity]@modres
 ]

Obligatory performance comparison:

tests = RandomInteger[1, 60, 100];

res = gifs[tests, 5, 4, 3]; // RepeatedTiming // First

0.000053

res == gridIndex[tests, 5, 4, 3]

True

gridIndex[tests, 5, 4, 3]; // RepeatedTiming // First

0.0064

getIndex[tests, 5, 4, 3]; // RepeatedTiming // First

0.00032

unrankList[5, 4, 3][tests]; // RepeatedTiming // First

0.00039

getInds[tests, 5, 4, 3]; // RepeatedTiming // First

0.000063

Clearly vectorization is doing what it should and getting us the performance we'd expect (which is interestingly better than a compiled implementation on my machine)

Here's a more detailed performance analysis which shows we're long-term a little bit better than Mathematica's auto-parallelization in compiled functions:

RandomSeed[123];
shape = RandomInteger[1, 10, 4];
sizes = 1, 5, 10, 50, 100, 500, 1000, 5000, 10000, 50000, 75000, 
 100000, 125000, 150000, 500000;
idxs = RandomInteger[1, Times @@ shape, #] & /@ sizes;

testC =
 MapThread[
 #, getInds[#2, shape]; // RepeatedTiming // First &,
 
 sizes,
 idxs
 
 ];

testU =
 MapThread[
 #, gifs[#2, shape]; // RepeatedTiming // First &,
 
 sizes,
 idxs
 
 ];

ListLinePlot[
 
 testC,
 testU
 ,
 PlotLegends -> "getInds", "gifs",
 PlotRange -> All
 ]