Here one way to plot many solutions for different C[1], 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]}]}
[![enter image description here][1]][1]


  [1]: https://i.sstatic.net/vBxkm.jpg