I love questions like this because they encourage me to experiment...
You CAN do this in MS Word 2003. I just fooled around a bit and found the answer. Here's how you do it.
First, make sure your Drawing Toolbar is active. To do this, go to Tools--> Customize. Make sure the "Tools" tab is selected and check the "Drawing" toolbar box if it isn't already checked. The Drawing Toolbar will appear at the bottom of your document window. OK your way out of the Customize dialog Box.
Now insert your picture. Insert--> Picture--> From file usually works best. If your picture is going to take up the entire page, FIRST hit Ctrl+Enter to make your document a two-page document, THEN insert the picture.
Now comes the delicious part, where we outwit MS Word...
Make sure you're on a blank page (the page you added above) or on a blank part of your page where the picture isn't. Make sure the picure isn't selected.
Now click the Text Box icon in the Drawing Tool Bar. If you don't know which it is, just hover your mouse pointer over each icon and it'll tell you what the icon does.
MS Word will add a "Drawing Area" with a grey border below your picture. Don't let it scare you. INSIDE this drawing area, click-drag to draw a rectangle. Type your text inside this box.
Once you're done typing, "grab" this text box by the thick, grey line around what you've typed and drag the whole box onto the picture. You can place it anywhere on the picture.
You'll find that the box is opaque, with black text on a white background. Nothing to worry about, we are going to change this.
Right-click the grey border and select, "Format text box" from the context menu that appears".
Make sure you're on the "Colors and Lines" tab.
In the "Fill" section, set Color to "No Fill" and in the "Line" section, set Color to "No Line".
Hit OK and you're done. Click-drag the text and change its colour so it stands out from your background picture, if required.