ListPointPlot3D with connected points?

One can use ListPointPlot3D and then replace Point with Line. By this way, the plot can enjoy options of ListPointPlot3D:

ListPointPlot3D[...]/. Point[a___] :> {Thick, Line[a]}

pts = RandomReal[{0, 1}, {19, 3}]; 
Show[ListPointPlot3D@pts, Graphics3D@Line@pts]

Edit:

Something slightly more useful as an example:

data = RandomFunction[RandomWalkProcess[0.3], {0, 10^2}, 3];
pts = Transpose@data["States"];
Show[ListPointPlot3D@#, Graphics3D@Line@#] &@pts

I was in the same situation, and what worked really well for me was this answer from a different question.

I'll start off with the random data generated by Dr belisarius:

data = RandomFunction[RandomWalkProcess[0.3], {0, 10^2}, 3];
pts = Transpose@data["States"];
ListPointPlot3D[pts, ColorFunction -> "SouthwestColors"]

pts is a list of points {{x1,y1,z1},{x2,y2,z2},...}, we convert this to a list of the form {{t1,{x1,y1,z1}},{t2,{x2,y2,z2}},...}:

ptsT = Transpose[{Range[1, Length@pts], pts}]

We then create a function to interpolate ptsT, which we can plot using ParametericPlot3D:

func = Interpolation[ptsT];
ParametricPlot3D[func[\[Tau]], {\[Tau], 1, Length@pts}, 
 ColorFunction -> 
  Function[{x, y, z, t}, ColorData["SouthwestColors"][z]]]