Right click on the ToolBar and select Edit Template
From Edit Template, select Edit a Copy
I recommend adding the copy to a Resource Dictionary
Click Ok
You'll now be editing the control template for the ToolBarPanel, and can set the visibility to Collapsed for the grip and overflow signal. You can rinse and repeat for the other controls. It is a bit time consuming, but isn't terribly hard with Blend.
The grip can be removed by setting the attached property ToolBarTray.IsLocked="True" on the ToolBar.
To remove the Overflow ToggleButton, you will have to remove it in a custom ControlTemplate as sixlettervariables suggests, which if you have blend or can download the Blend 3 Preview is not overly difficult.
You could also just hide the button in the loaded event of the ToolBar, though whichever route you take, you should also set the attached property ToolBar.OverflowMode="Never" on the ToolBar's menu, so that items cannot accidentally overflow into an unreachable area.
I am just starting out with WPF and could not get any of the above methods to hide my overflow arrow (Visual Studio 2010).The only thing that seemed to affect the arrow was the Toolbar_Load example above but all that did was turn the arrow into an empty space that looked as bad as the arrow. The easiest way I could figure was just to set the margins of the toolbar.
<ToolBar Height="26"
Name="toolBar"
DockPanel.Dock="Top"
ToolBarTray.IsLocked="True"
ToolBar.OverflowMode="Never" <!-- no effect -->
Margin="0,0,-13,0"> <!-- worked -->
<Menu ToolBar.OverflowMode="Never"> <!-- no affect -->
<MenuItem Header="_File"></MenuItem>
</Menu>
</ToolBar>
You can "remove" the overflow without supplying a new control template by setting the ToolBar to have negative right margins (and throw in a negative left margin so it doesn't look odd with rounded left edges but square right edges). Then, add ClipToBounds="True" to the ToolBarPanel which will cut off the edges of the toolbar which are now sticking outside the panel's area.
Rather than hiding the overflow button completely, I think it's better to show it only when necessary. This can be done by binding its Visibility property to its IsEnabled property: