Does the long-term ice-core temperature record provide reasonable evidence that the current temperature trend is not anthropogenic?
No, ice cores do not refute human-caused warming.  In fact, they support it.  Your logic really doesn't follow because the rate of warming now compared to some arbitrary previous date is not  related to whether the warming is driven by humans.  Instead, it is important to consider the natural variation of physical factors that affect climate (collectively known as Malenkovitch cycles; http://en.wikipedia.org/wiki/Milankovitch_cycles ), and then recognize any large deviations from those cycles.

An image of the EPICA and Vostok ice cores versus age obtained from http://en.wikipedia.org/wiki/Ice_age , clearly shows influence of the Malenkovitch cycles.  The most notable of which is the strong cycle that is approximately 70,000 years and is associated with geologic ice ages. In fact, the Earth is historically on average much cooler than it is now, and the retreat of glaciers has been periodic but not sustained.  We were supposed to have cooled off some time ago, but clearly we are not in a global cooling period. Many would even argue that humans started affecting the climate over 7,000 years ago when land management and fire was wielded prevalently.


In order to attribute climate change to humans, climate models are run with and without anthropogenic climate forcing.  When they do so, climate models run without anthropogenic forcing are unable to duplicate the current warming trend.  I've attached an image of this from the IPCC AR5 Technical Summary draft which shows a full climate simulation, one without anthropogenic climate forcers, and one with no aerosol forcing for comparison.



Finally, I thought someone might value this link that discusses climate change atrribution:
http://www.realclimate.org/index.php/archives/2014/08/ipcc-attribution-statements-redux-a-response-to-judith-curry/comment-page-4/