Plotting several functions

Do you mean something like this?

f[n_, x_] := Sin[x (1 + n/10)]
ParametricPlot3D[Evaluate[Table[{x, n, f[n, x]}, {n, 20}]], {x, 0, 2 Pi}]

Mesh will do the trick:

Plot3D[Sin[x^2 - y], {x, -2, 2}, {y, -3, 3}, MeshFunctions -> {#2 &}, 
 PlotStyle -> None, Mesh -> 30]

Placing the “wires” on integer values is also easy – see example below. For the range {y, -7, 5} there are 13 integers so you need to ask for 11 wires Mesh -> 11 (in red) because 2 are taken by boundary (blue). With such settings "wires" fall exactly on integer values.

Plot3D[Sin[x^2 - y/2], {x, -2, 2}, {y, -7, 5}, MeshFunctions -> {#2 &}, 
PlotStyle -> None, Mesh -> 11, MeshStyle -> {Thick, Red}, 
BoundaryStyle -> {Thick, Blue}]

I'm bored, so here is a recreation of Heike's plot without ParametricPlot3D

Graphics3D[
  Table[
    Cases[
      Plot[f[n, x], {x, 0, 2 Pi}],
      Line[x_] :> {ColorData[1][n], Line[{#,n,#2} & @@@ x]},
      {4}
    ],
    {n, 20}
  ],
  Axes -> True
]