The Cordiline's (Ti plants) will root but it all depends on the exposure to sunlight (not too much while in water) and the dryness of the air. These plants like it humid.

Or.. move to South Florida where we just stick them in the ground and wait!

What do you mean by "... the cutting is dry..."? That the cutting itself is dried out right through, or only on the surface? Dried right through would be dead. Generally they're very easy to strike, just need moisture and warmth. Although often put in water, they're better direct in moist soil. If the climate where you are is cold a heat mat would help.

And a week isn't very long for plants to start producing roots!

You can start your cuttings in potting soil with a bunch of perlite mixed in. Put them in the house under plant lights and you could give it a little bottom heat if you want it to root faster. Keep them damp but not soggy. Plant them horizontally in the plant media. I have rooted a number of them this way when I lived in a cold climate. Keep the larger plants away from drying heating vents in the winter and mist them periodically. They are not hard to grow. Indoors during the winter, they might get aphids. So rinse them off in the bathtub/shower if that happens. Only put them outside when the threat of frost is over.