Image3D can do this if we overcome two problems.
imgs = Import["https://i.stack.imgur.com/CQX0Q.gif"];
imgs = MapThread[SetAlphaChannel, {imgs, Binarize /@ imgs}]
Image3D[imgs, Background -> Black]
The second line makes all black in all the images transparent, this was the first problem.

The second problem is that the broccoli will be very flat. In order to fix this you may select the the image and click the "show options" button in the toolbar. If you change the voxel aspect ratio so that z is stretched five times compared to the x and y axis, you get this:

It is also possible to give it color:
colored = Colorize[#, ColorFunction -> ColorData["AvocadoColors"]] & /@ imgs;
imgs = MapThread[SetAlphaChannel, {colored, Binarize /@ imgs}];
Image3D[imgs, Background -> Black]

Animated:

Image3Dhttps://reference.wolfram.com/language/ref/Image3D.html – dr.blochwave Sep 26 '15 at 18:18