There is still some contention about whether leaf tissue density (mass per unit volume) leaf dry matter content (LDMC; dry mass per unit wet mass) are equivalent and whether past work has shown LDMC to be ecologically relevant, no less more relevant than specific leaf area (SLA).
I've looked through the literature pretty hard. Here's about all I can find:
1. LDMC and leaf tissue density should be positively correlated and there has been some excellent work investigating the underlying causes of variation in LDMC that are relevant for understanding leaf tissue density (Vile et al. 2005, Roderick et al. 1999, Shipley 1995). I still haven't found the perfect test of the two methods, but they should be pretty strongly related.
2. LDMC can predict plant strategies (Vendramini et al. 2002, Wilson et al. 1999). LDMC does a better job than SLA in predicting CSR placement for example.
3. LDMC can predict relative growth rates within species (Ryser and Aeschlimann 1999) and digestibility (Pontes et al. 2007, Ansquer et al. 2009, Duru et al. 2008).
4. LDMC was not correlated with competitive effect or response (Liancourt et al. 2009).
5. LDMC correlated better with soil fertility and sheep grazing intensity than SLA in Norwegian alpine ecosystems (Rusch et al.2009). [Note I still haven't read this paper--it's on order.]
That's about it.
The use of SLA still outnumbers tissue density or LDMC 50 to 1 and there still are essentially no published tests of the utility of either tissue density or LDMC in explaining abundance.