**Update 2:** Hiding the edges within the same community for general, not-necessarily tri-partite, graphs:

    ClearAll[insideEdges]
    insideEdges[g_, c_]:=Select[EdgeList[g], Or @@ (Function[c, SubsetQ[c, {##}] ]/@c)& @@ #&]

**Examples:**

    zkc = ExampleData[{"NetworkGraph", "ZacharyKarateClub"}];

    CommunityGraphPlot[zkc,  CommunityRegionStyle -> {LightRed, LightGreen, LightBlue},
     EdgeStyle -> {Alternatives @@ insideEdges[zkc, FindGraphCommunities[zkc]] :> Opacity[0]}]
[![enter image description here][1]][1]

    dsn= ExampleData[{"NetworkGraph", "DolphinSocialNetwork"}];

    CommunityGraphPlot[dsn,  CommunityRegionStyle -> 97,
     EdgeStyle -> {Alternatives @@ insideEdges[dsn, FindGraphCommunities[dsn]] :> Opacity[0]}]
[![enter image description here][2]][2]



**Update 1:**

    athreepartitegraph = CompleteGraph[{10, 7, 3}, 
      GraphLayout -> {"MultipartiteEmbedding",  "VertexPartition" -> {10, 7, 3}},
      ImageSize -> 400, VertexLabels -> "Name", ImagePadding -> 20];

    cgp = CommunityGraphPlot[athreepartitegraph, 
           {Range[10], Range[11, 17], Range[18, 20]}, Method -> "Hierarchical"];
    Row[{athreepartitegraph, cgp}] 
![enter image description here][3]

or

    CommunityGraphPlot[athreepartitegraph, 
      {Range[10], Range[11, 17], Range[18, 20]}, Method -> "SpringElectrical"]
![enter image description here][4]

to compare with alternative vertex layouts:

    Row[SetProperty[athreepartitegraph, 
        GraphLayout -> #] & /@ {{"CircularEmbedding"}, 
          {"CircularMultipartiteEmbedding", "VertexPartition" -> {10, 7, 3}}}]
![enter image description here][5]

---------------
**Original post:**

Maybe something like:

    g2 = ExampleData[{"NetworkGraph", "ZacharyKarateClub"}];
    g2 = SetProperty[g2, {VertexLabels -> "Name", ImagePadding -> 20, ImageSize -> 400}];

    fgp = FindGraphPartition[GraphComplement[g2], 3];
    mpg = Graph[Flatten@fgp, EdgeList[g2], 
       GraphLayout -> {"MultipartiteEmbedding", "VertexPartition" -> (Length /@ fgp)},
       VertexLabels -> "Name", 
       ImagePadding -> 20, ImageSize -> 400];

    Row[{g2, mpg}]
![enter image description here][6]

    CommunityGraphPlot[g2, fgp, Method -> "Hierarchical"]
![enter image description here][7]


  [1]: https://i.sstatic.net/LpBJS.png
  [2]: https://i.sstatic.net/avuio.png
  [3]: https://i.sstatic.net/xdThF.png
  [4]: https://i.sstatic.net/yNlcC.png
  [5]: https://i.sstatic.net/tVCOO.png
  [6]: https://i.sstatic.net/IrgNC.png
  [7]: https://i.sstatic.net/K2Q3D.png