There isn't a "PlotMarkers" for ListPointPlot3D.
One can disembowel the data structure for the graphic and change
Points to Text or graphics primitives, but that is a lot of work.
list1 = Table[Sin[j^2 + i], {i, 0, 3, 0.1}, {j, 0, 3, 0.1}];
list2 = Table[Sin[j + 2 i], {i, 0, 3, 0.1}, {j, 0, 3, 0.1}];
list3 = Table[Sin[j - i], {i, 0, 3, 0.1}, {j, 0, 3, 0.1}];
lpp3 = ListPointPlot3D[{list1, list2, list3}, PlotStyle -> PointSize[0.03]]
Look at the top of the internal data structure.
FullForm[lpp3]
Graphics3D[List[List[Directive[Hue[0.67`, 0.6`, 0.6`], PointSize[0.03`]],
Point[List[List[1.`, 1.`, 0.`], List[1.`, 2.`, 0.09983341664682815`],
List[1.`, 3.`, 0.19866933079506122`], ...
It would probably be easier to start with the lists of point coordinates, make them 3D,
list1 = Table[{i, j, Sin[j^2 + i]}, {i, 0, 3, 0.1}, {j, 0, 3, 0.1}];
list1 = Flatten[list1, 1]
create a function that makes pretty items at those coordinates, and use Graphics3D
to turn those into a plot.
mkPoint[{x_, y_, z_}] := {Green, Text["\[FilledDiamond]", {x, y, z}]}
Graphics3D[ Map[mkPoint, list1]]