It is definitely not a bug. There is only so much that you can do with dashing, so plot markers are useful for making monochrome curves distinguishable.
As rcollyer points out in a comment, the most straightforward way to get rid of the plot markers is to add that option along with PlotTheme:
ListLinePlot[{
Prime[Range[10]],
Fibonacci[Range[10]],
Range[10]
}, PlotTheme -> "Monochrome", PlotMarkers -> None]

Another option would be to create your own plot theme and use the setting given by Eldo to set the default plot theme to this.
As MichaelE2 showed here, and others elsewhere, there is a function called Charting`ResolvePlotTheme that will tell you exactly what the settings for a particular plot theme are.
style = Charting`ResolvePlotTheme["Monochrome", ListLinePlot];
ListLinePlot[{
Prime[Range[10]],
Fibonacci[Range[10]],
Range[10]
}, Sequence @@ style]

We can override options by prepending changes to style:
ListLinePlot[{
Prime[Range[10]],
Fibonacci[Range[10]],
Range[10]
}, Sequence @@ Prepend[style, PlotMarkers -> None]]

In other words, Prepend[style, PlotMarkers -> None] could be made its own plot theme and be used as the default plot theme.
Charting`ResolvePlotThemewhy don't you just usePlotMarkers -> Noneplus the theme? – rcollyer Jun 19 '17 at 13:53SetOptionsto setPlotMarkers -> Nonewhich would have the same effect. – rcollyer Jun 19 '17 at 14:27PlotMarkers -> Nonehas no effect, which is probably why Eldo didn't do that. I removed that paragraph now. – C. E. Jun 19 '17 at 15:13