BarChart3D with continuous data along one dimension

Plot3D example:

Plot3D[{
  2 Abs@Sin[x] Boole[0 < y < 2],
  2 Abs@Cos[x] Boole[2 < y < 4],
  4 Abs@Sin[x] Boole[4 < y < 6],
  4 Abs@Cos[x] Boole[6 < y < 8],
  6 Abs@Sin[x] Boole[8 < y < 10]
  },
 {x, 0, 2 Pi},
 {y, 0, 10},
 Filling -> Bottom,
 FillingStyle -> Opacity[1],
 Mesh -> False,
 PlotRange -> {{0, 2 Pi}, {0, 10}, {0, 10}}
 ]

Plot3D example

I'm assuming here that you already have the interpolation functions, and that what remains is plotting them.

If you need more control over the look than Plot3D gives you, the best way I can think of is to make the filling yourself with graphics primitives. Here's a proof of concept:

data = Table[{x, 0, Abs@Sin[x]}, {x, 0, 2 Pi, 0.01}];
left = Partition[data, 2, 1];
right = Partition[{0, 2, 0} + # & /@ data, 2, 1];

fillingPolygon[{{x1_, y1_, z1_}, {x2_, y2_, z2_}}] :=
 {
  EdgeForm[],
  ColorData["AvocadoColors", Mean[{z1, z2}]],
  Polygon[{
    {x1, y1, z1},
    {x2, y2, z2},
    {x2, y2, 0},
    {x1, y1, 0}
    }]
  }

Show[
 Graphics3D[{
   fillingPolygon /@ left,
   fillingPolygon /@ right
   }],
 Plot3D[
  Abs@Sin[x],
  {x, 0, 2 Pi},
  {y, 0, 2},
  Mesh -> False,
  ColorFunction -> "AvocadoColors",
  ColorFunctionScaling -> False
  ]
 ]

Custom filling example

Update: Combining all steps in a function:

ClearAll[minMax, hybridBarChart]
minMax = Table[f[[1]][{f[[2]][Through@#@x], #2[[1]] < x <= #2[[2]]}, 
       x] &[##], {f, {{NMaxValue, Max}, {NMinValue, Min}}}] &;

hybridBarChart[funcs_, range_, labels_: Automatic, cf_: "Rainbow"][
  opts : OptionsPattern[]] := Module[{minmax = minMax[funcs, range]},
  Show[Table[ParametricPlot3D[{t, i, v funcs[[i]][t]}, {t, range[[1]], 
      range[[2]]}, {v, 0, 1}, 
     ColorFunction -> (ColorData[{cf, minmax}][ funcs[[i]][#4]] &), 
     ColorFunctionScaling -> False, opts, Method -> "Extrusion" -> .5,
      PlotPoints -> 50, Mesh -> None], {i, Length@funcs}], 
   Ticks -> {Automatic, Transpose[{Range[Length@funcs], 
       labels /. Automatic -> Range[Length@funcs]}], Automatic}, 
   FilterRules[{opts}, Options @ Graphics3D], ImageSize -> Large, 
   Lighting -> "Neutral", PlotRange -> All, BoxRatios -> {1, 1, 1/2}]]

Examples :

functions = Function /@ (Range[5] Abs[Sin[#]]);

labels = "data" <> ToString[#] & /@ Range[Length @ functions];

hybridBarChart[functions, {0, 3 Pi}][ImageSize -> Medium, BaseStyle -> Opacity[.5]]

enter image description here

hybridBarChart[functions, {0, 3 Pi}, labels, "SolarColors"][
 Method -> "Extrusion" -> .8, ImageSize -> Medium]

enter image description here

Original answer:

You can also use ParametricPlot3D with the option "Extrusion":

functions = Function /@ (Range[5] Abs[Sin[#]]);

labels = "data" <> ToString[#] & /@ Range[Length @ functions];

max = NMaxValue[{Max[Through@functions@x], 0 < x <= 3 Pi}, x];
min = NMinValue[{Min[Through@functions@x], 0 < x <= 3 Pi}, x];

pp3D[f_, i_] := ParametricPlot3D[{t, i[[1]], v f[t]}, {t, 0, 3 Pi}, {v, 0, 1}, 
  PlotPoints -> 50, Mesh -> None, Method -> "Extrusion" -> .3, 
  ColorFunction -> (ColorData[{"Rainbow", {min, max}}][f[#4]] &), 
  ColorFunctionScaling -> False]

Show[MapIndexed[pp3D, functions],  ImageSize -> Large, 
 Lighting -> "Neutral", PlotRange -> All, BoxRatios -> {1, 1, 1/2}, 
 Ticks -> {Automatic, Transpose[{Range[Length@functions], labels}], Automatic}]

enter image description here