Yes, you can get beautiful hand dyed look with fabric inks and paints. I've been painting/inking on fabric for years. There are so many great products to use as well. My favorites are Setacolor (both transparent and opaque), Versatex Fabric Inks, Shiva paint sticks, Derwent Inktense pencils, and Luminier paints. Many other great ones out there but these are what I'm most familiar with so far. I also recently recieved Sharrill Kahn's new book "Creative Mixed Media" which has many fabulous examples of this and the techniques to get these results using paints, inks, colored pencils, crayons, etc. Great book and I highly recommend it if you want to learn a lot about painting on fabric and others ways to make some wonderful fabrics. I found my copy on amazon.com for a good price and you can look inside the book on their site.
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It depends what kind of hand dyed look you want. If you want the colours to blend into one another then make sure the fabric is wet so that the paints can run. Adding salt - ordinary, sea or rock - will also help to give you a mottled effect as the colour tries to escape from the salt and so you get an uneven finish.
Incidentally I have been using 'nasty dyes' for over 20 years and have never had any problem with them. Maggi