Here one way to plot many solutions for different C1, C[2], C[3].

dsol = DSolve[
         Thread[{x1'[t], x2'[t], 
                 x3'[t]} == {{-2, 1, 0}, {1, -2, 1}, {0, 1, -2}}.{x1[t], x2[t], 
                 x3[t]}], {x1, x2, x3}, t] // FullSimplify;

Since I work with MMA 8.0, I had to construct my own PlotLegend.

{ParametricPlot3D[
   Evaluate[({x1[t], x2[t], x3[t]} /. First@dsol) /. 
   Evaluate[
 tab = Flatten[
   Table[Thread[{C[1], C[2], C[3]} -> {i, j, k}], {i, -1, 
     1}, {j, -1, 1}, {k, -1, 1}], 2]]], {t, -1, 10}, 
    PlotStyle -> Table[Hue[.8 i/Length[tab]], {i, 1, Length[tab]}], 
   ImageSize -> 300], 
   Table[Graphics[{Hue[.8 i/Length[tab]], 
   Text[{C[1], C[2], C[3]} /. tab[[i]]]}, ImageSize -> 45], {i, 1, 
    Length[tab]}]}

Here one way to plot many solutions for different C1, C[2], C[3].

dsol = DSolve[
         Thread[{x1'[t], x2'[t], 
                 x3'[t]} == {{-2, 1, 0}, {1, -2, 1}, {0, 1, -2}}.{x1[t], x2[t], 
                 x3[t]}], {x1, x2, x3}, t] // FullSimplify;

Since I work with MMA 8.0, I had to construct my own PlotLegend.

{ParametricPlot3D[
   Evaluate[({x1[t], x2[t], x3[t]} /. First@dsol) /. 
   Evaluate[
 tab = Flatten[
   Table[Thread[{C[1], C[2], C[3]} -> {i, j, k}], {i, -1, 
     1}, {j, -1, 1}, {k, -1, 1}], 2]]], {t, -1, 10}, 
  PlotStyle -> 
  Table[Hue[.8 (i - 1)/Length[tab]], {i, 1, Length[tab]}], 
  ImageSize -> 500], 
  Table[Graphics[{Hue[.8 (i - 1)/Length[tab]], 
Text[{C[1], C[2], C[3]} /. tab[[i]]]}, ImageSize -> 45], {i, 1, 
Length[tab]}]}