They were as follows:
- tuberculosis (consumption), 2051
- diphtheria, 1117
- stillbirth, 909
- old age, 865
- pneumonia, 783
- heart disease, 668
- cholera infantum, 653
- apoplexy (stroke) & paralysis, 628
- typhoid fever, 498
- convulsions, 431
- cancer, 391
- casualty, 350
- dropsy (edema, excess fluid in the tissues), 333
- inflammation of brain, 332
- inflammation of bowels, 327
One thing that jumped out at me was that the third leading cause of death was stillbirth. I suppose this highlights the importance of prenatal care. It is also possible that part of the problem was women having very closely spaced pregnancies, which could compromise the health of the mother and hence her unborn child. According to the Mayo Clinic, closely spaced pregnancies may not allow the mother to recover sufficiently from a previous pregnancy or replace stores of essential nutrients. In addition, when a woman becomes pregnant within twelve months of giving birth it is even possible that the placenta partially or even completely separates from the uterus, which would obviously have dire consequences. Becoming pregnant within eighteen months of giving birth can also result in low birth weight or pre-term birth. 
The other interesting (and depressing) thing I found when I read a little more was the number of infants who died under one year of age. Of the roughly 40,000 births in the state in the 1886-1887 time period about 11% died before their first birthday. Even worse, about 42% of newborns who died in their first year of life actually died within a month of their birth.  This could also have something to do with closely spaced pregnancies for the reasons mentioned above. Obviously, there are many other possible reasons for these sad statistics, but it is beyond the scope of this little article to go into them here. Just as a comparison, in 1900 the infant mortality (children dying before their first birthday) was approximately 100 infant deaths per 1,000 live births.  By 2000, that rate had dropped to 6.89 deaths per 1,000 live births.
One final note belongs in the “the more things change the more they stay the same” category. If deaths from diseases preventable with antibiotics or vaccinations (excepting pneumonia) are removed from the 1886 list, the causes of death in the 1880s aren't that different from what is seen today.
1880s (in Michigan): stillbirth, old age, pneumonia, heart disease, stroke, convulsions, cancer, edema.
2009 (in the US): heart disease, cancer, lower respiratory diseases, stroke, casualty, Alzheimer's, diabetes, influenza/pneumonia, nephritis. 
Heart disease, stroke, cancer and pneumonia were killers then and unfortunately, remain so today. I wonder where things will stand in another 120 years or so.
- Twentieth Annual Report Relating to the Registry and Return of Births, Marriages and Deaths in Michigan for the Year 1886 by the Secretary of State of the State of Michigan. 1888. Thorp & Godfrey, State Printers and Binders. Lansing.
- Mayo Clinic Staff. Family Planning: Get the facts about pregnancy spacing. May 2011. http://www.mayoclinic.com/health/family-planning/MY01691.
- Twenty-First Annual Report Relating to the Registry and Return of Births, Marriages and Deaths in Michigan for the Year 1887 by the Secretary of State of the State of Michigan. 1889. Darius D. Thorp, State Printer and Binder. Lansing.
- MacDorman, M.F. and T.J. Mathews. Recent Trends in Infant Mortality in the United States. 2008. National Center for Health Statistics (NCHS) Brief Data Number 9. Hyattsville, MD
- Deaths: Final Data for 2009. January 2012. National Vital Statistics Report (NVSR) Volume 60, Number 3. National Center for Health Statistics. Hyattsville, MD.